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Problem 65

The scenes and the graph relate to the reaction of $\mathrm{X}_{2}(g)(\text {black})$ with $\mathrm{Y}_{2}(g)$ (orange) to form $\mathrm{XY}(g)$ . (a) If reactants and products are in their standard states, what quantity is represented on the graph by $x ?$ (b) Which scene represents point 1$?$ Explain. (c) Which scene represents point 2$?$ Explain.

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## Recommended Questions

The scenes below represent a chemical reaction between elements A (red) and B (green):

Which best represents the balanced equation for the reaction?

$\begin{array}{ll}{\text { (a) } 2 \mathrm{A}+2 \mathrm{B} \longrightarrow \mathrm{A}_{2}+\mathrm{B}_{2}} & {\text { (b) } \mathrm{A}_{2}+\mathrm{B}_{2} \longrightarrow 2 \mathrm{AB}} \\ {\text { (c) } \mathrm{B}_{2}+2 \mathrm{AB} \longrightarrow 2 \mathrm{B}_{2}+\mathrm{A}_{2}} & {\text { (d) } 4 \mathrm{A}_{2}+4 \mathrm{B}_{2} \longrightarrow 8 \mathrm{AB}}\end{array}$

The graph illustrates two chemical reactions. Which curve represents the reaction in the presence of the enzyme? What energy changes are represented by lines $a, b,$ and $c ?$

The scenes depict four initial reaction mixtures for the reaction of $A$ (blue) and $B$ (yellow), with and without a solid present (gray cubes). The initial rate, $-\Delta[\mathrm{A}] / \Delta t$ (in mol/L $\cdot$ s), is shown, with each sphere representing 0.010 mol and the container volume at 0.50 $\mathrm{L}$ .

(a) What is the rate law in the absence of a catalyst?

(b) What is the overall reaction order?

(c) Find the rate constant.

(d) Do the gray cubes have a catalytic effect? Explain.

This graph shows a plot of the rate of a reaction versus the

concentration of the reactant A for the reaction A->products.

a. What is the order of the reaction with respect to A?

b. Make a rough sketch of a plot of [A] versus time.

c. Write a rate law for the reaction including an estimate for the

value of k.

This graph shows a plot of the rate of a reaction versus the concentration of the reactant A for the reaction A $\longrightarrow$ products.

\begin{equation}\begin{array}{l}{\text { a. What is the order of the reaction with respect to A? }} \\ {\text { b. Make a rough sketch of a plot of [A] versus time. }} \\ {\text { c. Write a rate law for the reaction including an estimate for the }} \\ {\text { value of } k .}\end{array}\end{equation}

The diagram shown here represents a two-step mechanism. (a) Write the equation for each step and the overall reaction. (b) Identify the intermediate and catalyst. The color codes are $\mathrm{A}=$ green and $B=$ red.

This graph shows a plot of the rate of a reaction versus the concentration

of the reactant.

\begin{equation}

\begin{array}{l}{\text { a. What is the order of the reaction with respect to A? }} \\ {\text { b. Make a rough sketch of a plot of [A] versus time. }} \\ {\text { c. Write a rate law for the reaction including the value of } k \text { . }}\end{array}

\end{equation}

The accompanying graph shows plots of ln $k$ versus 1$/ T$ for two different reactions. The plots have been extrapolated to the $y$ -intercepts. Which reaction (red or blue) has

(a) the larger value for $E_{a},$ and (b) the larger value for the frequency factor, $A ?[$ Section 14.5$]$

This graph shows a plot of the rate of a reaction versus the

concentration of the reactant.

a. What is the order of the reaction with respect to A?

b. Make a rough sketch of a plot of [A] versus time.

c. Write a rate law for the reaction including the value of k.

Consider the following energy plots for a chemical reaction when answering the questions below.

a. Which plot (purple or blue) is the catalyzed pathway? How do you know?

b. What does $\Delta E_{1}$ represent?

c. What does $\Delta E_{2}$ represent?

d. Is the reaction endothermic or exothermic?

Consider the following energy plots for a chemical reaction when answering the questions below.

a. Which plot (purple or blue) is the catalyzed pathway? How do you know?

b. What does $\Delta E_{1}$ represent?

c. What does $\Delta E_{2}$ represent?

d. Is the reaction endothermic or exothermic?

Interpreting Graphics Given the activity table below, determine whether a reaction will occur or not. If the

reaction will occur, give the products.

a) L and $\mathrm{M}^{+}$

b) $\mathrm{P}$ and $\mathrm{M}^{+}$

c) $\mathrm{P}$ and $\mathrm{T}^{+}$

The molecular scenes below depict the reaction $\mathrm{Y} \Longrightarrow 2 \mathrm{Z}$ at four different times, out of sequence, as it reaches equilibrium. Each sphere (Y is red and $Z$ is green) represents 0.025 mol, and the volume is 0.40 $\mathrm{L}$ . (a) Which scene(s) represent(s) equilibrium? (b) List the scenes in the correct sequence. (c) Calculate $K_{\mathrm{c}}$ .

The equilibrium constant for the reaction $\mathrm{A} \rightleftarrows \mathrm{B}$ is $K_{\mathrm{c}}=$ 10 at a certain temperature. (1) Starting with only reactant A, which of the diagrams shown here best represents the system at equilibrium? (2) Which of the diagrams best represents the system at equilibrium if $K_{c}=0.10 ?$ Explain why you can calculate $K_{\mathrm{c}}$ in each case without knowing the volume of the container. The grey spheres represent the A molecules, and the green spheres represent the B molecules.

Study the graph below, and answer the questions that follow. For help in interpreting graphs, see Appendix $B,$ "Study Skills for Chemistry."

The graph relates to an experiment in which the concentrations of bromide ion, hydrogen peroxide, and bromine were monitored as the following reaction took place.

$$\begin{array}{c}{2 \mathrm{Br}^{-}(a q)+\mathrm{H}_{2} \mathrm{O}_{2}(a q)+2 \mathrm{H}_{3} \mathrm{O}^{+}(a q) \rightarrow} \\ {\quad \mathrm{Br}_{2}(a q)+4 \mathrm{H}_{2} \mathrm{O}(l)}\end{array}$$

Associate each curve with one of the species being monitored.

For the reaction depicted in the molecular scenes, X is red and Y is green.

(a) Write a balanced equation.

(b) Determine the sign of $\Delta S_{\mathrm{rxn}}$

(c) Which species has the highest molar entropy?

Elements $X(\text {green})$ and $\mathrm{Y}$ (purple) react according to the following equation: $\mathrm{X}_{2}+3 \mathrm{Y}_{2} \longrightarrow 2 \mathrm{XY}_{3} .$ Which molecular

scene represents the product of the reaction?

An equilibrium mixture of two solids and a gas, in the reaction $\mathrm{XY}(s) \rightleftharpoons$ $\mathrm{X}(g)+\mathrm{Y}(s),$ is depicted at right $(\mathrm{X} \text { is }$ green and $\mathrm{Y}$ is black). Does scene A, B, or C best represent the system at equilibrium after two formula units of $\mathrm{Y}(s)$ is added? Explain.

Consider the following graph of the total free energy of reactants and products versus reaction progress for a general reaction, Reactants $\longrightarrow$ Products:

(a) Account for the shape of the curve, and identify the point at which $\Delta G=0 .$ What is the significance of that point?

(b) Why is the minimum in the plot on the left side of the graph?

The reaction of A with B to give D proceeds in two steps and can be represented by the following Hess’s law diagram.

(a) What is the equation and $\Delta H^{\circ}$ for the net reaction?

(b) Which arrow on the diagram corresponds to which step, and which arrow corresponds to the net reaction?

(c) The diagram shows three energy levels. The energies of which substances are represented by each?

The following diagram represents an imaginary two-step mechanism. Let the red spheres represent element A, the green ones element $B$ , and the blue ones element $\mathrm{C}$ . (a)

Write the equation for the net reaction that is occurring. (b) Identify the intermediate. (c) Identify the catalyst. [Sections 14.6 and 14.7$]$

Study the graph below, and answer the questions that follow. For help in interpreting graphs, see Appendix $B,$ "Study Skills for Chemistry."

The graph relates to an experiment in which the concentrations of bromide ion, hydrogen peroxide, and bromine were monitored as the following reaction took place.

$$\begin{array}{c}{2 \mathrm{Br}^{-}(a q)+\mathrm{H}_{2} \mathrm{O}_{2}(a q)+2 \mathrm{H}_{3} \mathrm{O}^{+}(a q) \rightarrow} \\ {\quad \mathrm{Br}_{2}(a q)+4 \mathrm{H}_{2} \mathrm{O}(l)}\end{array}$$

Measure the slope of each of the three curves at $t=500 \mathrm{s}.$

The following molecular scenes depict the aqueous reaction $2 \mathrm{D} \rightleftharpoons \mathrm{E},$ with $\mathrm{D}$ red and $\mathrm{E}$ blue. Each sphere represents 0.0100 $\mathrm{mol}$ , but the volume is 1.00 $\mathrm{L}$ in scene $\mathrm{A},$ whereas in scenes $\mathrm{B}$ and $\mathrm{C},$ it is 0.500 $\mathrm{L}$

(a) If the reaction in scene $\mathrm{A}$ is at equilibrium, calculate $K_{\mathrm{c}} .$

(b) Are the reactions in scenes $\mathrm{B}$ and $\mathrm{C}$ at equilibrium? Which, if either, is not, and in which direction will it proceed?

The reaction between reactant A (blue spheres) and reactant $\mathrm{B}($ red spheres) is shown in the following diagram:

Based on this diagram, which equation best describes the reaction? [ Section 3.1$]$

$$

\begin{array}{l}{\text { (a) } \mathrm{A}_{2}+\mathrm{B} \longrightarrow \mathrm{A}_{2} \mathrm{B}} \\ {\text { (b) } \mathrm{A}_{2}+4 \mathrm{B} \longrightarrow 2 \mathrm{AB}_{2}} \\ {\text { (c) } 2 \mathrm{A}+\mathrm{B}_{4} \longrightarrow 2 \mathrm{AB}_{2}} \\ {\text { (d) } \mathrm{A}+\mathrm{B}_{2} \longrightarrow \mathrm{AB}_{2}}\end{array}

$$

The scenes below depict three gaseous mixtures in which $\mathrm{X}$

(orange) and $\mathrm{Y}_{2}$ (black) are reacting to form $\mathrm{XY}$ and $\mathrm{Y}$ . Assume that each gas has a partial pressure of 0.10 $\mathrm{atm} .$

(a) If $K=4.5$ , which mixture is at equilibrium? (b) Rank the mixtures from the most positive $\Delta G$ to the most negative $\Delta G .$

Figure 18.33 shows how energy changes during the

progress of a reaction.

\begin{equation}

\begin{array}{l}{\text { a. Is the reaction exothermic or endothermic? }} \\ {\text { (Chapter } 15 )} \\ {\text { b. How many steps are in the reaction mechanism for }} \\ {\text { the reaction? (Chapter } 16 )} \\ {\text { c. Explain how you could use the graph to identify the }} \\ {\text { rate-determining step. (Chapter } 16 )}\end{array}

\end{equation}

Study the graph below, and answer the questions that follow. For help in interpreting graphs, see Appendix $B,$ "Study Skills for Chemistry."

What relationship does the graph describe?

The equilibrium constant for the reaction $\mathrm{A} \rightleftharpoons \mathrm{B}$ is $K_{\mathrm{c}}=10$ at a certain temperature. (1) Starting with only reactant $\mathrm{A}$ which of the diagrams shown here best represents the system at equilibrium? (2) Which of the diagrams best represents the system at equilibrium if $K_{c}=0.10 ?$ Explain why you can calculate $K_{\mathrm{c}}$ in each case without knowing the volume of the container. The gray spheres represent the $\mathrm{A}$ molecules and the green spheres represent the $\mathrm{B}$ molecules.

Explain what each reaction below shows. Describe the relationship between the amounts of products and reactants for each case.

a. reactants $\rightleftarrows$ products

b. reactants $\longleftrightarrow$ products

The plot below shows the number of collisions with a particular energy for two different temperatures.

a. Which is greater, $T_{2}$ or $T_{1} ?$ How can you tell?

b. What does this plot tell us about the temperature dependence of the rate of a chemical reaction? Explain your answer.

Study the graph below, and answer the questions that follow. For help in interpreting graphs, see Appendix $B,$ "Study Skills for Chemistry."

Miles measures the energy required to hold two magnets apart at varying distances. He notices that it takes less and less energy to hold the magnets apart as the distance between them increases. Compare the results of Miles's experiment with the data given in the graph.

Use these two graphs to answer the following questions.

GRAPH

a. Briefly explain which point on the Phillips curve graph best represents the same economic situation as point $\mathrm{B}$ on the aggregate demand and aggregate supply graph.

b. Briefly graph. griefly explain which point on the Phillips curve graph best represents the same economic situation as point $C$ on the aggregate demand and aggregate sup- ply graph.

For the reaction $\mathrm{M}_{2}+\mathrm{N}_{2} \rightleftharpoons 2 \mathrm{MN}$ , scene A represents the mixture at equilibrium, with M black and N orange. If each molecule represents 0.10 mol and the volume is 1.0 $\mathrm{L}$ , how many moles of each substance will be present in scene $\mathrm{B}$ when that mixture reaches equilibrium?

A friend studies a first-order reaction and obtains the following three graphs for experiments done at two different temperatures. (a) Which two graphs represent experiments done at the same temperature? What accounts for the difference in these two graphs? In what

way are they the same? (b) Which two graphs represent experiments done with the same starting concentration but at different temperatures? Which graph probably represents the lower temperature? How do you know? [Section 14.4$]$

Demonstrate The table below shows the concentrations of Substances $A$ and $B$

in two reaction mixtures. $A$ and $B$ react according to the equation 2$A \rightleftharpoons B; $ $K_{\mathrm{eq}}=200$ . Are the two mixtures at different equilibrium positions?

(A) Sketch the graph of $x^{2}+y^{2}=4$ by hand and identify the curve.

(B) Graph $y_{1}=\sqrt{4-x^{2}}$ and $y_{2}=-\sqrt{4-x^{2}}$ in the standard viewing window of a graphing calculator. How do these graphs compare to the graph you drew in part A?

(C) Apply each of the following ZOOM options to the graphs in part $B$ and determine which options produce a curve that looks like the curve you drew in part A:

ZDecimal, ZSquare, ZoomFit.

The following scenes represent a gaseous reaction between compounds of nitrogen ($blue$) and oxygen ($red$) at 298 K:

(a) Write a balanced equation and use Appendix $\mathrm{B}$ to calculate $\Delta H_{\mathrm{rxx}}^{\mathrm{o}}$

(b) If each molecule of product represents $1.50 \times 10^{-2} \mathrm{mol}$ , what quantity of heat (in $\mathrm{J} )$ is released or absorbed?

Match each graph (labeled $a$, $b, c,$ and $d$ ) with the appropriate equation.

(GRAPHS CAN'T COPY)

$$\frac{x^{2}}{9}+y^{2}=1$$

Match each graph (labeled $a, b, c,$ and $d$ ) with the appropriate equation.

(GRAPHS CAN'T COPY)

$$y^{2}=3 x$$

Graphical Analysis (a) use a graphing utility to graph the equation, (b) use the graph to approximate any $x$ -intercepts of the graph, (c) set $y=0$ and solve the resulting equation, and (d) compare the result of part (c) with the $x$ -intercepts of the graph.

$$y=\frac{1}{x}-\frac{4}{x-1}-1$$

Use the graph below to answer the following questions.

a. Which of the points $A, B, C,$ or $D$ can represent a long- run equilibrium?

b. Suppose that initially the economy is at point A. If aggregate demand increases from $A D_{1}$ to $A D_{2}$ which point represents short-run equilibrium? Which point represents the eventual long-run equilibrium? Briefly explain how the economy adjusts from the short-run equilibrium to the long-run equilibrium.

(a) For the generic reaction $\mathrm{A} \rightarrow \mathrm{B}$ what quantity, when graphed versus time, will yield a straight line for a first-order reaction? (b) How can you calculate the rate

constant for a first-order reaction from the graph you made in part (a)?

Which of the following graphs of Gibbs energy versus the extent of reaction represents an equilibrium constant closest to $1 ?$

The plot below shows the number of collisions with a particular energy for two different temperatures.

a. Which is greater, $T_{2}$ or $T_{1} ?$ How can you tell?

b. What does this plot tell us about the temperature of the rate of a chemical reaction? Explain your answer.

Scenes A, B, and C below depict the following reaction at three temperatures:

$$\mathrm{NH}_{4} \mathrm{Cl}(s) \rightleftharpoons \mathrm{NH}_{3}(g)+\mathrm{HCl}(g) \quad \Delta H_{\mathrm{mn}}^{\circ}=176 \mathrm{kJ}$$

(a) Which best represents the reaction mixture at the highest temperature? Explain. (b) Which best represents the reaction mixture at the lowest temperature? Explain.

Use the following graph for problems 1.53 and 1.54:

a. What is measured on the horizontal axis?

b. What is the range of values on the horizontal axis?

c. What is the solubility of carbon dioxide in water at $25^{\circ} \mathrm{C} ?$

d. At what temperature does carbon dioxide have a solubility of $0.2 \mathrm{g} / 100 \mathrm{g}$ water?

match the equation with its graph. [The graphs are labeled (a), (b), (c), and (d).

$y=\sqrt{9-x^{2}}$

The reaction $\mathrm{A}_{2}(g)+\mathrm{B}(g) \rightleftharpoons \mathrm{A}(g)+\mathrm{AB}(g)$ has an equilibrium constant of $K_{p}=2 .$ The accompanying diagram shows a mixture containing A atoms (red), $\mathrm{A}_{2}$ molecules, and $A B$ molecules (red and blue). How many B atoms should be added to the diagram to illustrate an equilibrium mixture? [Section 15.6$]$

Match each graph (labeled $a, b, c,$ and $d$ ) with the appropriate equation.

(GRAPHS CAN'T COPY)

$$x^{2}=-2 y$$

Interpreting Graphs The graphs in the two viewing windows shown here can be used to solve the equation $3 \sqrt{x}+4=x^{2}-1$ graphically.

(a) The viewing window in (a) illustrates the intersection method for solving. Identify the two equations that are graphed.

(b) The viewing window in (b) illustrates the x-intercept method for solving. Identify the equation that is graphed.

(c) Writing to Learn How are the intersection points in (a) related to the x-intercepts in (b)?

The following diagrams represent a hypothetical reaction $\mathrm{A} \longrightarrow \mathrm{B},$ with $\mathrm{A}$ represented by red spheres and $\mathrm{B}$ represented by blue spheres. The sequence from left to right represents the system as time passes. Does the system reach equilibrium? If so, in which diagram(s) is the system in equilibrium? [Sections 15.1 and 15.2]

Consider the reaction $\mathrm{N}_{2} \mathrm{O}(g)-\mathrm{Au}, \mathrm{N}_{2}(g)+\frac{1}{2} \mathrm{O}_{2}(g)$

(a) Does the gold catalyst (Au, above the arrow) act as a homogeneous or a heterogeneous catalyst?

(b) On the same set of axes, sketch the reaction energy diagrams for the catalyzed and the uncatalyzed reaction.

Describe and explain how the concentrations of $\mathrm{A}, \mathrm{B}$

$\mathrm{C},$ and D change from the time when $\mathrm{A}$ and $\mathrm{B}$ are first

combined to the point at which equilibrium is established for the reaction $\mathrm{A}+\mathrm{B} \rightleftarrows \mathrm{C}+\mathrm{D}$

Match each graph (labeled $a$, $b, c,$ and $d$ ) with the appropriate equation.

(GRAPHS CAN'T COPY)

$$x^{2}+\frac{y^{2}}{4}=1$$

The “filmstrip” represents five molecular scenes of a gaseous mixture as it reaches equilibrium over time:

$\mathrm{X}$ is purple and $\mathrm{Y}$ is orange: $\mathrm{X}_{2}(g)+\mathrm{Y}_{2}(g) \rightleftharpoons 2 \mathrm{XY}(g)$

(a) Write the reaction quotient, $Q,$ for this reaction.

(b) If each particle represents $0.1 \mathrm{mol},$ find $Q$ for each scene.

(c) If $K>1,$ is time progressing to the right or to the left? Explain.

(d) Calculate $K$ at this temperature.

(e) If $\Delta H_{\mathrm{rxn}}^{\circ}<0,$ which scene, if any, best represents the mixture at a higher temperature? Explain.

(f) Which scene, if any, best represents the mixture at a higher pressure (lower volume)? Explain.

Use the following graph for problems 1.53 and 1.54:

a. What does the title indicate about the graph?

b. What is measured on the vertical axis?

c. What is the range of values on the vertical axis?

d. Does the solubility of carton dioxide increase or decrease with an increase in temperature?

(a) use a graphing utility to graph the two equations in the same viewing window and (b) use the table feature of the graphing utility to create a table of values for each equation. (c) What do the graphs and tables suggest? Verify your conclusion algebraically.$$\begin{array}{l}y_{1}=\ln \left[x^{2}(x-4)\right] \\y_{2}=2 \ln x+\ln (x-4)\end{array}$$

The following diagrams represent the progress of the reaction $\mathrm{A} \longrightarrow \mathrm{B},$ where the red spheres represent $\mathrm{A}$ molecules and the green spheres represent B molecules. Calculate the rate constant of the reaction.

(A) Sketch the graph of $x^{2}+y^{2}=9$ by hand and identify the curve.*

(B) Graph $y_{1}=\sqrt{9-x^{2}}$ and $y_{2}=-\sqrt{9-x^{2}}$ in the standard viewing window of a graphing calculator. How do these graphs compare to the graph you drew in part A?

(C) Apply each of the following ZOOM options to the graphs in part $B$ and determine which options produce a curve that looks like the curve you drew in part A:

ZDecimal, ZSquare, ZoomFit.

Consider an initial mixture of $\mathrm{N}_{2}$ and $\mathrm{H}_{2}$ gases that can be represented as follows:The gases react to form ammonia gas $\left(\mathrm{NH}_{3}\right)$ as represented by the following concentration profile:a. Label each plot on the graph as $\mathrm{N}_{2}, \mathrm{H}_{2},$ or $\mathrm{NH}_{3},$ and explain your answers.

b. Explain the relative shapes of the plots.

c. When is equilibrium reached? How do you know?

The reaction $\mathrm{A}_{2}+\mathrm{B}_{2} \rightleftharpoons 2 \mathrm{AB}$ has an equilibrium constant $K_{c}=1.5 .$ The following diagrams represent reaction mixtures containing $\mathrm{A}_{2}$ molecules (red), $\mathrm{B}_{2}$ molecules (blue), and AB molecules, (a) Which reaction mixture is at equilibrium? (b) For those mixtures that are not at equilibrium, how will the reaction proceed to reach equilibrium? [Sections 15.5 and 15.6$]$

Graphical Analysis (a) use a graphing utility to graph the equation, (b) use the graph to approximate any $x$ -intercepts of the graph, (c) set $y=0$ and solve the resulting equation, and (d) compare the result of part (c) with the $x$ -intercepts of the graph.

$$y=|x+1|-2$$

Consider the data presented in Exercise 14.19 . (a) By using appropriate graphs, determine whether the reaction is first order or second order. (b) What is the rate constant for the reaction? (c) What is the half-life for the reaction?

Graph the functions. Then answer the following questions.

a. How does the graph behave as $x \rightarrow 0^{+} ?$

b. How does the graph behave as $x \rightarrow \pm \infty ?$

c. How does the graph behave near $x=1$ and $x=-1 ?$

Give reasons for your answers.

$$y=\frac{3}{2}\left(x-\frac{1}{x}\right)^{2 / 3}$$

Graph the functions. Then answer the following questions.

a. How does the graph behave as $x \rightarrow 0^{+} ?$

b. How does the graph behave as $x \rightarrow \pm \infty ?$

c. How does the graph behave near $x=1$ and $x=-1 ?$

Give reasons for your answers.

$$y=\frac{3}{2}\left(\frac{x}{x-1}\right)^{2 / 3}$$

Graphical Analysis (a) use a graphing utility to graph the equation, (b) use the graph to approximate any $x$ -intercepts of the graph, (c) set $y=0$ and solve the resulting equation, and (d) compare the result of part (c) with the $x$ -intercepts of the graph.

$$y=|x-2|-3$$

Study the graph below, and answer the questions that follow. For help in interpreting graphs, see Appendix $B,$ "Study Skills for Chemistry."

What do the blue spheres represent on this graph?

Graphical Analysis (a) use a graphing utility to graph the equation, (b) use the graph to approximate any $x$ -intercepts of the graph, (c) set $y=0$ and solve the resulting equation, and (d) compare the result of part (c) with the $x$ -intercepts of the graph.

$$y=\sqrt{7 x+36}-\sqrt{5 x+16}-2$$

(a) Graph the two functions $y=x^{2}+4 x$ and $y=x^{2}-4 x$ How are the two graphs related (in terms of symmetry)?

(b) Follow part (a) using the two functions $y=-2 x^{2}+3 x+4$ and $y=-2 x^{2}-3 x+4$

(c) Which one of the graphing techniques from Section 3.4 relates to what you have observed in parts (a) and (b)?

Discuss, and illustrate with a graph, how each of the following events, ceteris paribus, will affect the market for coffee:

a. A blight on coffee plants kills off much of the Brazilian crop.

b. The price of tea declines.

c. Coffee workers organize themselves into a union and gain higher wages.

d. Coffee is shown to cause cancer in laboratory rats.

e. Coffee prices are expected to rise rapidly in the near future.

a. Sketch the graph of $y=x^{2}$

b. Sketch the graph of $y=3 x^{2}$

c. Sketch the graph of $y=\frac{1}{3} x^{2}$

d. Describe the graph of $y=a x^{2}$ in terms of the graph of $y=x^{2}$ when $a > 1$

e. Describe the graph of $y=a x^{2}$ in terms of the graph of $y=x^{2}$ when $0 < a < 1$

Match the equation with its graph. [The graphs are labeled (a), (b), (c), and (d).]

$$\text { } \frac{(y-2)^{2}}{4}-\frac{(x-2)^{2}}{9}=1$$

a. Identify the equilibrium values. Which are stable and which are unstable?

b. Construct a phase line. Identify the signs of $y^{\prime}$ and $y^{\prime \prime}$

c. Sketch several solution curves.

$$y^{\prime}=y^{3}-y^{2}$$

Consider an initial mixture of $\mathrm{N}_{2}$ and $\mathrm{H}_{2}$ gases that can be represented as follows:

The gases react to form ammonia gas $\left(\mathrm{NH}_{3}\right)$ as represented by the following concentration profile:

a. Label each plot on the graph as $\mathrm{N}_{2}, \mathrm{H}_{2},$ or $\mathrm{NH}_{3},$ and

explain your answers.

b. Explain the relative shapes of the plots.

c. When is equilibrium reached? How do you know?

Match each equation with its graph. The graphs are labeled ( $a$ ) through $(d)$.

(Check your book to see graph)

$y=x^{2}-x-2$

Match the equation with its graph. [The graphs are labeled (a), (b), (c), and (d).]

$$\text { } \frac{y^{2}}{a}-\frac{x^{2}}{a}=1$$

Consider the reaction: $\mathrm{A}(g) \rightleftharpoons 2 \mathrm{B}(g) .$ The graph plots the

concentrations of $\mathrm{A}$ and $\mathrm{B}$ as a function of time at a constant temperature. What is the equilibrium constant for this reaction at this temperature?

Graph the following equations and explain why they are not graphs of functions of $x$

a. $|x|+|y|=1$

b. $|x+y|=1$

Match each graph (labeled $a, b, c,$ and $d$ ) with the appropriate equation.

(GRAPHS CAN'T COPY)

$$y=3 x^{2}$$

Graphical Analysis (a) use a graphing utility to graph the equation, (b) use the graph to approximate any $x$ -intercepts of the graph, (c) set $y=0$ and solve the resulting equation, and (d) compare the result of part (c) with the $x$ -intercepts of the graph.

$$y=x-5+\frac{7}{x+3}$$

Graphical Analysis (a) use a graphing utility to graph the equation, (b) use the graph to approximate any $x$ -intercepts of the graph, (c) set $y=0$ and solve the resulting equation, and (d) compare the result of part (c) with the $x$ -intercepts of the graph.

$$y=x^{4}-10 x^{2}+9$$

Graphical Analysis (a) use a graphing utility to graph the equation, (b) use the graph to approximate any $x$ -intercepts of the graph, (c) set $y=0$ and solve the resulting equation, and (d) compare the result of part (c) with the $x$ -intercepts of the graph.

$$y=x^{4}-29 x^{2}+100$$

Describe the effect temperature has on the rate of the reactions using the graph above.

CAN'T COPY THE GRAPH

Match cach graph (labeled $a$, $b, c,$ and $d$ ) with the appropriate equation.

GRAPH CANT COPY

$$\frac{x^{2}}{9}-y^{2}=1$$

Match each graph (labeled $a$, $b, c,$ and $d$ ) with the appropriate equation.

(GRAPHS CAN'T COPY)

$$\frac{(x-1)^{2}}{9}+\frac{(y-1)^{2}}{4}=1$$

Graphical Analysis (a) use a graphing utility to graph the equation, (b) use the graph to approximate any $x$ -intercepts of the graph, (c) set $y=0$ and solve the resulting equation, and (d) compare the result of part (c) with the $x$ -intercepts of the graph.

$$y=2 x+\frac{8}{x-5}-2$$

Match cach graph (labeled $a$, $b, c,$ and $d$ ) with the appropriate equation.

GRAPH CANT COPY

$$y^{2}-\frac{x^{2}}{9}=1$$

Match each graph (labeled $a, b, c,$ and $d$ ) with the appropriate equation.

(GRAPHS CAN'T COPY)

$$y^{2}=-\frac{x}{2}$$

In these exercises we compare the graphs of two exponential functions

a. Sketch the graphs of $f(x)=9^{x / 2}$ and $g(x)=3^{x}$

b. Use the Laws of Exponents to explain the relationship between these graphs.

Match the equations listed in parts (a)-(d) to the graphs in the accompanying figure.

a. $y=(x-1)^{2}-4$

b. $y=(x-2)^{2}+2$

c. $y=(x+2)^{2}+2$

d. $y=(x+3)^{2}-2$

(Check your book to see graph)

Match the equation with its graph. [The graphs are labeled (a), (b), (c), and (d).]

$$\text { } \frac{x^{2}}{4}-\frac{(y+3)^{2}}{0}=1$$

The images below represent the reactants of

a chemical reaction. Study the images, then

answer the items that follow.

$$\begin{array}{l}{\text { a. Write a balanced chemical equation for }} \\ {\text { the reaction that shows the states of all }} \\ {\text { substances. }} \\ {\text { b. What type of reaction is this? }}\end{array}$$

Use the following graph for questions 1.23 and 1.24:

a. What is measured on the horizontal axis?

b. What is the range of values on the horizontal axis?

c. What is the temperature of the tea after 20 min?

d. How many minutes were needed to reach a temperature of $45^{\circ} \mathrm{C} ?$

Sketch the graph of $y=\cos b x$ for $b=\frac{1}{2}, 2$ and $3 .$ How does the value of $b$ affect the graph? How many complete cycles of the graph of $y$ occur between 0 and $2 \pi$ for each value of $b ?$

Match each equation with its graph. Explain your choices.(Don't use a computer or graphing calculator.)

$\begin{array}{ll}{\text { (a) } y=3 x} & {\text { (b) } y=3^{x}} \\ {\text { (c) } y=x^{3}} & {\text { (d) } y=\sqrt[3]{x}}\end{array}$

The scene below represents a mixture of A2 and B2 before they react to form $\mathrm{AB}_{3}$

(a) What is the limiting reactant?

(b) How many molecules of product can form?

match the equation with its graph. [The graphs are labeled (a), (b), (c), and (d).

$y=x^{3}-x$

Draw a graph showing the reaction pathway for an overall exothermic reaction with two intermediates that are produced at different rates. On your graph indicate the reactants, products, intermediates, transition states, and activation energies. [Sections 14.6 and 14.7$]$

The diagram shows the energy of a reaction as the reaction progresses.

Label each blank box in the diagram.

\begin{equation}

\begin{array}{l}{\text { a. reactants }} \\ {\text { b. products }} \\ {\text { c. activation energy }\left(E_{2}\right)} \\ {\text { d. enthalpy of reaction }\left(\Delta H_{\mathrm{rm}}\right)}\end{array}

\end{equation}